The polarization vectors define the unpolarized photon density matrix
\[\sum_{h=\pm}\epsilon _\mu(\vec k, h)\epsilon^*_\nu(\vec k, h)=-g_{\mu\nu}+{k_\mu\bar k_\nu+\bar k_\mu k_\nu\over k\cdot\bar k}\ .
\]
It is easy to verify, using the identity \(x\delta'(x)=-\delta(x)\) and \(x\delta(x)=0\), that for a generic choice of the functions \(a\ ,\alpha\ ,\ \beta\) the above equations give the general solution to the Landau's gauge free field equations.